Interlacing and asymptotic properties of Stieltjes polynomials

TL;DR

This paper investigates the zero distribution and interlacing properties of Stieltjes polynomials, deriving new asymptotic behaviors and establishing that these polynomials do not form orthogonal sequences.

Contribution

It introduces two new interlacing theorems for Stieltjes polynomials and their zeros, and explores their asymptotic properties and non-orthogonality.

Findings

Zeros of successive Stieltjes polynomials of the same degree interlace

Zeros of Stieltjes polynomials of successive degrees interlace

No sequence of Stieltjes polynomials is orthogonal

Abstract

Polynomial solutions to the generalized Lam\'e equation, the \textit{Stieltjes polynomials}, and the associated \textit{Van Vleck polynomials} have been studied since the 1830's, beginning with Lam\'e in his studies of the Laplace equation on an ellipsoid, and in an ever widening variety of applications since. In this paper we show how the zeros of Stieltjes polynomials are distributed and present two new interlacing theorems. We arrange the Stieltjes polynomials according to their Van Vleck zeros and show, firstly, that the zeros of successive Stieltjes polynomials of the same degree interlace, and secondly, that the zeros of Stieltjes polynomials of successive degrees interlace. We use these results to deduce new asymptotic properties of Stieltjes and Van Vleck polynomials. We also show that no sequence of Stieltjes polynomials is orthogonal.